Noise sensitivity of the minimum spanning tree of the complete graph

We study the noise sensitivity of the minimum spanning tree (MST) of the <![CDATA[ $n$ ]]> -vertex complete graph when edges are assigned independent random weights. It is known that when the graph distance is rescaled by <![CDATA[ $n^{1/3}$ ]]> and vertices are given a uniform measure, the MST converges in distribution in the Gromov-Hausdorff-Prokhorov (GHP) topology. We prove that if the weight of each edge is resampled independently with probability <![CDATA[ $\varepsilon \gg n^{-1/3}$ ]]>, then the pair of rescaled minimum spanning trees - before and after the noise - converges in distribution to independent random spaces. Conversely, if <![CDATA[ $\varepsilon \ll n^{-1/3}$ ]]>, the GHP distance between the rescaled trees goes to <![CDATA[ $0$ ]]> in probability. This implies the noise sensitivity and stability for every property of the MST that corresponds to a continuity set of the random limit. The noise threshold of <![CDATA[ $n^{-1/3}$ ]]> coincides with the critical window of the Erd 's-Rényi random graphs. In fact, these results follow from an analog theorem we prove regarding the minimum spanning forest of critical random graphs.